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We define a large class of multifractal random measures and processes with arbitrary log-infinitely divisible exact or asymptotic scaling law. These processes generalize within a unified framework both the recently defined log-normal Multifractal Random Walk processes (MRW) [33, 3] and the log-Poisson " product of cynlindrical pulses " [7]. Their… (More)

- E Bacry, S Delattre, M Hoffmann, J F Muzy
- 2011

We prove a law of large numbers and a functional central limit theorem for multivariate Hawkes processes observed over a time interval [0, T ] in the limit T → ∞. We further exhibit the asymptotic behaviour of the covariation of the increments of the components of a multivariate Hawkes process, when the observations are imposed by a discrete scheme with… (More)

We give general mathematical results concerning oscillating singularities and we study examples of functions composed only of oscillating singularities. These functions are deened by explicit coeecients on an orthonormal wavelet basis. We compute their HH older regularity and oscillation at every point and we deduce their spectrum of oscillating… (More)

In this paper we discuss the problem of the estimation of extreme event occurrence probability for data drawn from some multifractal process. We also study the heavy (power-law) tail behavior of probability density function associated with such data. We show that because of strong correlations, the standard extreme value approach is not valid and classical… (More)

In this paper, we provide a simple, " generic " interpretation of multifractal scaling laws and multiplicative cascade process paradigms in terms of volatility correlations. We show that in this context 1/f power spectra, as recently observed in Ref. [20], naturally emerge. We then propose a simple solvable " stochastic volatility " model for return… (More)

- E Bacry, S Delattre, M Hoffmann, J F Muzy
- 2011

We introduce a new stochastic model for the variations of asset prices at the tick-by-tick level in dimension 1 (for a single asset) and 2 (for a pair of assets). The construction is based on marked point processes and relies on mutually exciting stochastic intensities as introduced by Hawkes. We associate a counting process with the positive and negative… (More)

Singular behavior of functions are generally characterized by their HH older exponent. However, we show that this exponent poorly characterizes oscillating singularities. We thus introduce a second exponent that that accounts for the oscillations of a singular behavior and we give a characterization of this exponent using the wavelet transform. We then… (More)

- E Bacry, S Delattre, M Hoffmann, J F Muzy
- 2010

We introduce a new stochastic model for the variations of asset prices at the tick-by-tick level in dimension 1 (for a single asset) and 2 (for a pair of assets). The construction is based on marked point processes and relies on linear self and mutually exciting stochastic intensities as introduced by Hawkes. We associate a counting process with the… (More)

- E Bacry, A Kozhemyak, J F Muzy
- 2008

Log-normal continuous random cascades form a class of multifractal processes that has already been successfully used in various fields. Several statistical issues related to this model are studied. We first make a quick but extensive review of their main properties and show that most of these properties can be analytically studied. We then develop an… (More)

In this paper we revisit an idea originally proposed by Mandelbrot about the possibility to observe " negative dimensions " in random multifractals. For that purpose, we define a new way to study scaling where the observation scale ℓ and the total sample length L are respectively going to zero and to infinity. This " mixed " asymptotic regime is… (More)